# Erf/gaussian integral

I'm working on algorithm that identifies clusters of objects together in space. One of the calculations in the algorithm is of the form

$$P = \int_0^\infty e^{-ax^2}{\rm erf}\left(\frac{x}{c} + b\right)dx$$

I am seeking an analytical solution to this integral, either in the defining bounds $$(0,\infty)$$, or in an approximation using bounds $$(0,10)$$, where I know that $$x=10$$ happens to be a good approximation for $$x\rightarrow\infty$$ given the behavior of the integrand.

Closed form for definite integral involving Erf and Gaussian?

Integral of a gaussian times ("shifted") erf squared

but these have different bounds and are of a different forms (i.e., less constants). Mathematica was not able to solve the integral, but my attempt has been to differentiate under the integral sign:

$$\frac{dP}{db} = \frac{2}{\sqrt{\pi}} e^{-b^2} \int_0^\infty e^{-2bx/c}e^{-(a+1/c^2)x^2}dx.$$

I was able to reduce this to

$$\frac{dP}{db} = \frac{1}{\sqrt{a+1/c^2}} e^{-b^2\left(1-\frac{1}{ac^2+1}\right)} {\rm erfc}\left[ \frac{b}{\sqrt{ac^2+1}} \right].$$

(pending any mistakes along the way, which is definitely possible). The problem is that this integral needed to get $$P(b)$$ is essentially the same as the one I started with. Do you all have any recommendations on how to proceed from this step, or better suggestions overall to solve this problem?

• One of the constants $a$ or $c$ can be removed.
– user65203
Nov 15, 2019 at 18:07
• @YvesDaoust do you mean by writing a in terms of c?
– zh1
Nov 15, 2019 at 18:21
• Nov 15, 2019 at 18:58
• Rescaling of the variable.
– user65203
Nov 15, 2019 at 19:36